This coincides with the definition given in [1]}.
For the calculations below, we will need some more details about this matrix representation.
Let \(E_{I}=(e_{i},\tilde{e}^{i})\)  be a basis for \(L\oplus \tilde{L}\) , with \(e_{i}\)  and \(\tilde{e}^{i}\)  bases for
\(L\)  and \(\tilde{L}\)  respectively. Any vector \(X\in L\oplus \tilde{L}\)  can then be decomposed as
\(X=x^{i}e_{i}+\tilde{x}_{i}\tilde{e}^{i}\) .
Let \(F^{I}=(\theta ^{i},\tilde{\theta }_{j})\)  be the dual basis, so that \(F^{I}(E_{J})=\delta ^{I}_{J}\) .
The map \(K_{t}\in {\rm Aut}(L\oplus \tilde{L})\)  can be expanded as
\(K_{t}=(K_{t})^{I}{}_{J}E_{I}\otimes F^{J}\) , and the matrix representation (REF ) means that
\(K_{t} = e_{i}\otimes \theta ^{i} - \tilde{e}^{i}\otimes \tilde{\theta }_{i} +2(B_{t})_{ij}\tilde{e}^{i}\otimes \theta ^{j}.\) 
